11.2 continuous probability distributions

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CHAPTER 3

23

Continuous Probability

Distributions

n this chapter, we define continuous probability distributions anddescribe the most popular ones. We look closely at the normal probabil-

ity distribution, which is the mainstay of finance theory, risk measures,and performance measures—and its appeal, despite the preponderance of empirical evidence and theoretical arguments that many financial eco-nomic variables do not follow a normal distribution. We postpone until

Chapter 7 a discussion of a class of continuous probability distributions,called stable Paretian distributions, which we show in later chapters aremore appropriate in many applications in finance.

CONTINUOUS RANDOM VARIABLES ANDPROBABILITY DISTRIBUTIONS

If the random variable can take on any possible value within the range of outcomes, then the probability distribution is said to be a continuous ran-dom variable.1 When a random variable is either the price of or the returnon a financial asset or an interest rate, the random variable is assumed tobe continuous. This means that it is possible to obtain, for example, aprice of 95.43231 or 109.34872 and any value in between. In practice,we know that financial assets are not quoted in such a way. Nevertheless,there is no loss in describing the random variable as continuous and in

1 Precisely, not every random variable taking its values in a subinterval of the realnumbers is continuous. The exact definition requires the existence of a density func-tion such as the one that we use later in this chapter to calculate probabilities.

I



Continuous Probability Distributions 29

The cumulative distribution function is another way to uniquely charac-terize an arbitrary probability distribution on the set of real numbersand in Chapter 7 we learn a third possibility, the so-called characteristicfunction.

THE NORMAL DISTRIBUTION

The class of normal distributions is certainly one of the most importantprobability distributions in statistics and, due to some of its appealingproperties, the class that is used in most applications in finance. Here weintroduce some of its basic properties.

Panel a of Exhibit 3.3 shows the density function of a normal distri-bution with μ = 0 and σ = 1. A normal distribution with these parametervalues is called a standard normal distribution. Notice the following

P X t ≤( ) F t ( ) f x( ) xd

∞ –

t

∫ = =

EXHIBIT 3.3 Density Function of a Normal DistributionPanel a: Standard Normal Distribution (μ = 0, σ = 1)



32 PROBABILITY AND STATISTICS

pendently normally distributed with μ = 0.05% and σ = 1.6%. Then themonthly returns again are normally distributed with parameters μ =1.05% and σ = 7.33% (assuming 21 trading days per month) and theyearly return is normally distributed with parameters μ = 12.6% and σ= 25.40% (assuming 252 trading days per year). This means that theS&P 500 monthly return fluctuates randomly around 1.05% and theyearly return around 12.6%.

The last important property that is often misinterpreted to justify thenearly exclusive use of normal distributions in financial modeling is thefact that the normal distribution possesses a domain of attraction. A

mathematical result called the central limit theorem states that—undercertain technical conditions—the distribution of a large sum of randomvariables behaves necessarily like a normal distribution. In the eyes of many, the normal distribution is the unique class of probability distribu-tions having this property. This is wrong and actually it is the class of sta-ble distributions (containing the normal distributions), which is unique inthe sense that a large sum of random variables can only converge to a sta-ble distribution. We discuss the stable distribution in Chapter 7.

OTHER POPULAR DISTRIBUTIONS

In the remainder of this chapter, we provide a brief introduction to somepopular distributions that are of interest for financial applications andwhich might be used later in this book. We start our discussion with theexponential distribution and explain the concept of hazard rate. thisleads to the class of Weibull distributions, which can be interpreted asgeneralized exponential distributions. Together with the subsequentlyintroduced class of Chi square distributions, the Weibull distributionsbelong to the more general class of Gamma distributions. We continueour exposition with the Beta distribution, which can be of particularinterest for credit risk modelling and the t distribution. The log-normaldistribution is the classical and most popular distribution when model-ing stock price movements. Subsequently, we introduce the logistic and

extreme value distribution. The latter has become popular in the field of operational risk analysis and is contained in the class of generalizedextreme value distributions. We conclude the chapter with a short dis-cussion of the generalized Pareto and the skewed normal distributionand the definition of a mixed distribution.5

5 For a thorough treatment of all mentioned distributions, the reader is referred tothe standard reference Johnson, Kotz, and Balakrishnan (1994 and 1995).



46 PROBABILITY AND STATISTICS

functions f 1,…f n and n positive real numbers α1,…αn with the propertyΣαi = 1 and define a new probability density f via

The so defined mixed distributions are often used when no well-knowndistribution family seems appropriate to explain the specific observedphenomenon.

REFERENCES

Embrechts, P., C. Klueppelberg, and T. Mikosch. 1997. Modelling Extremal Eventsfor Insurance and Finance, vol. 33 of Applications of Mathematics. Berlin: Springer-Verlag.

Johnson, N. L., S. Kotz, and N. Balakrishnan. 1994. Continuous Univariate Distri-bution, Volume 1, 2nd ed . New York: John Wiley & Sons.

Johnson, N. L., S. Kotz, and N. Balakrishnan. 1995. Continuous Univariate Distri-bution, Volume 2, 2nd ed . New York: John Wiley & Sons.

f x( ) αif i x( )i 1=

n

∑=

11.2 continuous probability distributions

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